CN1695151A - CAD system and CAD program - Google Patents

Abstract

A CAD system and a CAD program capable of significantly increasing the utility value of a CAD model and efficiency of the design/production process by employing the theory of surfaces guaranteeing the continuity of a free curve/curved surface. The program causes a computer to execute a point sequence extraction processing for extracting a plurality of point sequences on a curved surface, a division processing for generating a curved surface from a point sequence by using another CAD system and dividing the curved surface into a predetermined number of meshes, a primary standard amount calculation processing for calculating a primary standard amount defined by a tangent vector forming a mesh tangent plane, a secondary standard amount calculation processing for calculating a secondary standard amount defined by a tangent vector and a mesh normal vector, and a storage processing for storing the point sequence information, the first standard amount, and the second standard amount.

Description

Computer aided design system and program

Technical Field

The present invention relates to a Computer Aided Design (CAD) system and a CAD program for deforming the shape of a component into a curved surface shape of a target.

Background

In order to meet the demands of consumers, it is desired to shorten the process from planning to designing and production. Cg (computer graphics) and cad (computer aid design) systems are widely used to improve the efficiency of the design and production processes. In order to express a shape having a complicated curved line or curved surface shape such as an automobile and a home electric product on a computer, the following processing method has been conventionally used.

The first is a solid model (solid model) in which a simple shape called an "original shape (native)" is retained in a computer, and an operation of combining these shapes is repeated to express a complicated shape. The original shape is, for example, a cylinder, a cube, a rectangular parallelepiped, a torus, a sphere, or the like, and the shape is expressed by a collective operation of these original shapes in the three-dimensional model. Therefore, many steps are required to form a complicated shape, and rigorous calculation is required.

The second type is a surface model, in which an operation of cutting and connecting lines or planes is performed by using an algorithm such as bezier, b-spline, Rational bezier, NURBS (Non-Uniform Rational b-spline), and a complicated free curve or curved surface is expressed by the repetition of the operations.

However, even if the three-dimensional model and the surface model have no problem in terms of expression, problems may occur when the models are used in downstream applications such as CAM and CAE. This is because the model is corrected by modifying the application of a translator or the like, such as the difference between the support element for CG support and the support elements for CG, CAD, and downstream application support, or the difference in shape definition, which is created, for example (japanese patent application laid-open nos. 2001-250130, 11-65628, 10-69506, 4-134571, 4-117572, and 1-65628).

However, the above-described correction work is extremely inefficient in achieving a reduction in design and production processes. The reason why correction is necessary varies depending on the individual cases, but is particularly problematic in the production process because all curves and curved surfaces are approximated by euclidean geometry in the conventional CG and CAD systems. For example, in the case of generating the list cylinder of the saddle shape shown in fig. 6 through a scanning (sweep) operation, there are a long line of the edge portion of the saddle and a short line of the center portion of the saddle. Thus, the scanning operation is a deformation accompanying the expansion and contraction of the pattern in order to maintain the continuity of the generated curved surface. However, in the conventional CG and CAD systems, the expansion and contraction are not considered, and the internal appearance is approximately cylindrical. Therefore, when such a CG model or CAD model that is approximately represented by euclidean geometry is actually transferred to CAE, an error generated therein becomes a problem in production.

Disclosure of Invention

The present invention has been made to solve the above-described problems, and an object of the present invention is to provide a CAD system and a CAD program that can significantly improve the use value of a CG model or a CAD model and can efficiently perform designing and manufacturing processes.

The CAD system of the invention is characterized by comprising: a point row information extraction section that extracts a plurality of point rows on the curved surface; a dividing means for generating a curved surface from the point sequence by using another CAD system and dividing the curved surface into a predetermined number of meshes; a primary specification amount calculation unit that calculates a primary specification amount defined by a tangent vector of a tangent plane forming the mesh; a secondary specification amount calculation unit that calculates a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and a storage unit that stores the dot row information, the primary specification amount, and the secondary specification amount.

Furthermore, the CAD system of the invention is characterized by further comprising: a principal curvature calculation unit that calculates a principal curvature of the mesh based on the primary and secondary specification amounts; a curvature line calculation unit that calculates a curvature line representing a principal direction of the mesh based on the principal curvature; a feature point/feature line analyzing unit that extracts a point or a line serving as a reference point or a reference line of a deformation defined by a variation pattern of one or two or more feature amounts from five feature amounts indicating a feature of the curved surface, the five feature amounts being composed of a gaussian curvature and an average curvature calculated based on the principal curvature, the principal direction, the curvature line, and the primary and secondary specification amounts; a perimeter (girth) length calculating section that calculates a perimeter based on the curvature calculated from the primary and secondary specification amounts.

The CAD system according to the present invention further includes a reproduction unit that reproduces the mesh or the curved surface by deforming only the curvature line of the circumferential portion in the curvature line direction with the feature point or the feature line as a reference for deformation.

Further, the CAD system of the invention is characterized by further comprising a conversion means for extracting a plurality of point sequences on a curved surface from the reproduced mesh or curved surface and converting the point sequences according to a graphic representation algorithm in another CAD system.

The CAD program of the present invention is characterized by causing a computer to execute: extracting point column information, namely extracting a plurality of point columns on the curved surface; a dividing process of generating a curved surface by using another CAD system based on the point sequence and dividing the curved surface into a predetermined number of meshes; a primary specification amount calculation process of calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh; a secondary specification amount calculation process of calculating a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and a storing process of storing the dot column information, the primary specification amount, and the secondary specification amount.

Further, the CAD program of the invention also causes the computer to execute: a principal curvature calculation process of calculating a principal curvature of the mesh based on the primary and secondary specification amounts; a curvature line calculation process of calculating a curvature line representing a principal direction of the mesh from the principal curvature; a feature point/feature line analysis process of extracting a point or a line serving as a reference point or a reference line of a deformation defined by a variation pattern of one or two or more feature amounts, from five feature amounts representing features of the curved surface, the five feature amounts being composed of the principal curvature, the principal direction, the curvature line, and the gaussian curvature and the average curvature calculated from the primary and secondary specification amounts; and a perimeter calculation process of calculating a perimeter based on the curvature calculated from the primary and secondary specification amounts.

The CAD program of the present invention causes a computer to execute a reproduction process of deforming only the curvature line of the circumferential portion in the curvature line direction with the feature point or the feature line as a reference for deformation, and reproducing the mesh or the curved surface.

Further, the CAD program of the present invention causes a computer to execute a conversion process of extracting a plurality of point sequences on a curved surface from the reproduced mesh or curved surface and converting the point sequences according to a graphic representation algorithm in another CAD system.

The CG system of the present invention is characterized by comprising: a point row information extraction section that extracts a plurality of point rows on the curved surface; a dividing means for generating a curved surface from the point sequence by using another CG system and dividing the curved surface into a predetermined number of meshes; a primary specification amount calculation unit for calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh; a secondary specification amount calculation unit that calculates a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and a storage unit that stores the dot row information, the primary specification amount, and the secondary specification amount.

The CG program of the present invention causes a computer to execute: extracting point column information, namely extracting a plurality of point columns on the curved surface; a division process of generating a curved surface using another CG system from the point array and dividing the curved surface into a predetermined number of meshes; a primary specification amount calculation process of calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh; a secondary specification amount calculation process of calculating a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and a storing process of storing the dot column information, the primary specification amount, and the secondary specification amount.

The present invention has the following effects.

Due to the fact that the method comprises the following steps: a point row information extraction section that extracts a plurality of point rows on the curved surface; a dividing means for generating a curved surface from the point sequence by using another CG or CAD system and dividing the curved surface into a predetermined number of meshes; a primary specification amount calculation unit that calculates a primary specification amount defined by a tangent vector of a tangent plane forming the mesh; a secondary specification amount calculation unit that calculates a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and a storage unit for storing the point array information, the primary specification amount, and the secondary specification amount, so that the utilization value of the CG model or the CAD model can be greatly improved and the design and production can be efficiently performed by using a curved surface theory for ensuring the continuity of a free curve or a curved surface.

Moreover, since it includes: a principal curvature calculation unit that calculates a principal curvature of the mesh based on the primary and secondary specification amounts; a curvature line calculation unit that calculates a curvature line representing a principal direction of the mesh based on the principal curvature; a feature point/feature line analyzing unit that extracts a point or a line serving as a reference point or a reference line of a deformation defined by a variation pattern of one or two or more feature amounts from five feature amounts indicating a feature of the curved surface, the five feature amounts being composed of a gaussian curvature and an average curvature calculated based on the principal curvature, the principal direction, the curvature line, and the primary and secondary specification amounts; the perimeter calculation means calculates the perimeter based on the curvature calculated from the primary and secondary specifications, and therefore can reproduce and convert a CG or CAD model analyzed by the surface theory into another CG or CAD model.

Further, since the computer program further includes a reproduction means for reproducing the mesh or the curved surface by deforming only the curvature line of the peripheral portion in the curvature line direction with the feature point or the feature line as a reference for deformation, the CG or CAD model analyzed by the curved surface theory can be reproduced.

Further, since the conversion means for extracting a plurality of point sequences on the curved surface from the reproduced mesh or curved surface and converting the point sequences in accordance with the graphic expression algorithm in another CG or CAD system is further included, the CG or CAD model which is theoretically analyzed from the curved surface can be converted into another CG or CAD model.

Drawings

Fig. 1 is a configuration diagram showing a configuration of a CAD system according to the present embodiment.

Fig. 2 is an explanatory diagram showing a case where a curved surface is divided into m × n meshes and the base vectors Su and Sv are defined.

Fig. 3 is an explanatory view showing a plane of the unit tangent vector t and the unit normal vector n.

Fig. 4 is a flowchart showing a flow of processing from the free-form surface analysis by the analysis program 1 to data transfer.

Fig. 5 is an explanatory diagram showing a case where the curvature changes.

Fig. 6 is an explanatory diagram showing classification of the mean curvature and the gaussian curvature.

Fig. 7 is an explanatory view showing an equi-oblique orthogonal line.

Fig. 8 is an explanatory diagram showing a principal curvature extremum line.

Fig. 9 is an explanatory diagram showing a case where the extreme tilt line and the gaussian curvature distribution are inclined.

Fig. 10 is an explanatory view showing a curvature line.

Detailed Description

An embodiment of the CAD system of the invention is explained below with reference to the drawings. Fig. 1 is a configuration diagram showing a configuration of a CAD system according to the present embodiment. The CAD system according to the present embodiment is configured by a central processing unit (not shown) such as a CPU, a memory (not shown) such as a ROM and a RAM, a database 10, an image display processing unit 11, a display unit 12, an output unit 13, and a communication unit (not shown).

The CPU reads the analysis program 1, the conversion program 2, and the reproduction program 3 stored in the ROM, and executes a series of processes related to analysis, conversion, and reproduction of the free-form surface. The RAM is a semiconductor memory for temporarily storing data by the CPU.

The analysis program 1 reads actual measurement data 20 of a three-dimensional object such as CAT and other CAD format data 21 (for example, graphic data expressed by a three-dimensional model and surface models such as bezier, b-spline, rational bezier, NURBS), creates a point list information table 30, a primary specification table 31, and a secondary specification table 32, and executes processing stored in the database 10 by the CPU.

As shown in FIG. 2, the dot column information table 30 is composed of

And point column information (u, v) on the curved surface expressed in a parametric form of s (u, v) { x (u, v), y (u, v), z (u, v) }0 ≦ u, v ≦ 1 … … (equation 1). For example, when u is 0, 1/m, 2/m, … … m-1/m (m is a natural number), and v is 0, 1/n, 2/n … … n-1/n (n is a natural number), the curved surface shown in fig. 2 is divided into m × n meshes. In this case, the dot sequence information (u, v) is mn data strings of the grid IDs 1 to IDmn.

The primary specification gauge 31 is constituted by a primary specification E, F, G derived from the following equation. When u and V are functionally related, s (u, V) represents a curve on a curved surface, s/Su represents a tangent vector of the curve where u is constant, and s/V represents a tangent vector of the curve where V is constant. At this time, the basis vectors Su, Sv form tangent planes to the curved surface. Furthermore, a vector ds connecting s (u + du, v + dv) from two points s (u, v) on the curved surface is represented by

ds＝s_udu+s_vdv … … (equation 2)

And (4) showing. Here, the square of the absolute value of ds is represented by

(ds)²＝ds·ds＝s_u ²(du)²+2s_u·s_vdudv+s_v ²(dv)²… … (equation 3) shows that the above-mentioned primary specification amount is defined by the following equation based on the basis vector of the curved surface.

E＝s_u ²，F＝s_u·s_v，G＝s_v ²… … (equation 4)

The primary specification amount E, F, G is uniquely identified in each grid as described above, and the primary specification amount table 31 stores values for grid IDs 1 to IDmn.

The above equations 3 and 4 are summarized as

ds²＝E(du)²+2Fdudv+G(dv)²… … (equation 5).

The secondary specification gauge 32 is constituted by a secondary specification L, M, N derived by the following equation. When the angle formed by the base vectors Su and Sv is ω, the absolute value H of the vector product of the inner product F and the base vector is expressed as follows using a primary standard value.

<math> <mrow> <mi>F</mi> <mo>=</mo> <mo>|</mo> <msub> <mi>s</mi> <mi>u</mi> </msub> <mo>|</mo> <mo>&CenterDot;</mo> <mo>|</mo> <msub> <mi>s</mi> <mi>v</mi> </msub> <mo>|</mo> <mi>cos</mi> <mi>&omega;</mi> <mo>=</mo> <mrow> <mo>(</mo> <msqrt> <mi>EG</mi> </msqrt> <mo>)</mo> </mrow> <mi>cos</mi> <mi>&omega;</mi> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 6)

H＝|s_u×s_v|＝|S_u|·|s_v|sinω

<math> <mrow> <mo>=</mo> <msqrt> <mi>EG</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>cos</mi> <mn>2</mn> </msup> <mi>&omega;</mi> <mo>)</mo> </mrow> </msqrt> <mo>=</mo> <msqrt> <mi>EG</mi> <mo>-</mo> <msup> <mi>F</mi> <mn>2</mn> </msup> </msqrt> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 7)

Using this calculated value H, the unit normal vector n on the curved surface is expressed by the following equation.

<math> <mrow> <mi>n</mi> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mi>Su</mi> <mo>&times;</mo> <mi>Sv</mi> <mo>)</mo> </mrow> <mi>H</mi> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 8)

As shown in fig. 3, a line bundle of tangent vectors at a point P on the curved surface exists in the tangent plane, and one unit tangent vector t is expressed by the following equation.

$t=\frac{\mathrm{ds}}{\mathrm{ds}}=\mathrm{Su}\left(\frac{\mathrm{du}}{\mathrm{ds}}\right)+\mathrm{Sv}\left(\frac{\mathrm{dv}}{\mathrm{ds}}\right)......$ (equation 9)

As shown in fig. 3, the plane defined by t and n is referred to as a normal plane.

The curvature κ of the point P on the normal section is called normal curvature, and when t is differentiated from the arc length s of the normal section, it becomes

$\frac{\mathrm{dt}}{\mathrm{ds}}=\mathrm{Su}\frac{d^{2}u}{d{s}^2}+\mathrm{Sv}\frac{d^2\mathrm{<figure-callout\; id="2"\; label="v\; d\; s"\; filenames="A20038010076500171.png,A20038010076500211.png"\; state="\{\{state\}\}">v</figure-callout>}}{d{s}^{}}+\mathrm{Suu}{\left(\frac{\mathrm{du}}{\mathrm{ds}}\right)}^2+2\mathrm{Suv}\left(\frac{\mathrm{du}}{\mathrm{ds}}\right)\left(\frac{\mathrm{dv}}{\mathrm{ds}}\right)+\mathrm{Svv}{\left(\frac{\mathrm{dv}}{\mathrm{ds}}\right)}^2......$ (equation 10).

The normal vector is multiplied on both sides, and the following quadratic specification amount is introduced

L＝n·s_uu，M＝n·s_uv，N＝n·s_w… … (equation 11) is

<math> <mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>&CenterDot;</mo> <mi>n</mi> <mo>)</mo> </mrow> <mi>&kappa;</mi> <mo>=</mo> <mi>L</mi> <msup> <mrow> <mo>(</mo> <mfrac> <mi>du</mi> <mi>ds</mi> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>M</mi> <mrow> <mo>(</mo> <mfrac> <mi>du</mi> <mi>ds</mi> </mfrac> <mo>)</mo> </mrow> <mo>(</mo> <mfrac> <mi>dv</mi> <mi>ds</mi> </mfrac> <mo>)</mo> <mo>+</mo> <mi>N</mi> <msup> <mrow> <mo>(</mo> <mfrac> <mi>dv</mi> <mi>ds</mi> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 12).

The secondary specification amount L, M, N is uniquely identified in each grid as described above, and the secondary specification amount table 32 stores values for grid IDs 1 to IDmn.

When formula 5 is substituted for formula 12, the following formula is obtained.

<math> <mrow> <mi>&kappa;</mi> <mo>=</mo> <mfrac> <mrow> <mi>L</mi> <msup> <mrow> <mo>(</mo> <mi>du</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>Mdudv</mi> <mo>+</mo> <mi>N</mi> <msup> <mrow> <mo>(</mo> <mi>dv</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mi>E</mi> <msup> <mrow> <mo>(</mo> <mi>du</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>Fdudv</mi> <mo>+</mo> <mi>G</mi> <msup> <mrow> <mo>(</mo> <mi>dv</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 13)

The primary specification amount and the secondary specification amount are calculated from the above.

The conversion program 2 reads information necessary for the free-form surface from the point list information table 30, the primary specification table 31, and the secondary specification table 32, generates free-form surface data, and causes the computer to execute processing for transforming the free-form surface data into a shape that can be interpreted by another CAD application.

The playback program 3 reads information necessary for the free-form surface from the dot column information table 30, the primary specification table 31, and the secondary specification table 32, generates free-form surface data, and causes the computer to execute processing for output to the image display processing unit 11, as in the case of the conversion program 2.

The database 10 stores the dot column information table 30, the primary specification table 31, and the secondary specification table 32, and the output result of the analysis program 1 is written in association with a grid ID described later.

The image display processing unit 11 executes image display processing of output results from a playback program and other CAD applications.

The display unit 12 displays the output result of the image display processing unit 11.

The output unit 13 outputs the output result of the image display processing unit 11 to a communication unit or another recording medium. The communication unit transmits data such as the dot sequence information, the primary standard amount, and the secondary standard amount stored in the database 1 to other servers and clients via a network such as a LAN or the internet.

Next, a flow of a series of processes of free-form surface analysis, conversion, and reproduction in the CAD system according to the present embodiment will be described with reference to the drawings. Fig. 4 is a flowchart showing a flow of processing from the free-form surface analysis to data transfer in the analysis program 1.

The CPU receives an analysis command of the actually measured value data 20 and other CAD format data 21 by a user operation, reads the analysis program 1 from the ROM, and executes a free-form surface analysis process. First, the CPU performs a process of extracting a plurality of dot rows on a curved surface such as a two-dimensional NURBS surface and a double three-dimensional curved surface held by the actually measured value data 20 and the other CAD format data 21. Then, a curved surface is generated from the point sequence by using another CAD system (step S1 in fig. 4), and as shown in fig. 2, the curved surface is divided into a predetermined number mn of meshes, and then each mesh portion is normalized by the basic vectors Su and Sv. The point list information (u, v) generated at the time of normalization and the grid ID are simultaneously associated with the point list information table 30 held by the database 10 and written.

Next, the CPU executes differential geometry analysis processing. That is, the process of calculating the primary specification amount E, F, G defined by the base vectors Su, Sv of the tangent planes forming the mesh is executed. The calculated primary specification amount E, F, G is written in association with the primary specification amount table 31 held in the database 10 at the same time as the grid ID, as with the dot column information. Then, the CPU performs a process of calculating a quadratic specification L, M, N defined by the basis vectors Su and Sv and the unit normal vector n of the grid. The calculated secondary gauge amount L, M, N is written in association with the secondary gauge table 32 held by the database 10 at the same time as the grid ID, similarly to the primary gauge amount E, F, G.

The CPU performs a process of calculating an integratable condition which is a condition that a differential equation representing the mesh is continuous at a boundary surface of each mesh, in other words, a condition that the differential equation has a uniform solution.

Now, the above-described curved surface coordinates (u, v) and (u1, u2) are replaced, and this point is set to p (u1, u 2). When a curve in which u2 can be fixed and u1 can be moved is referred to as a u1 curve and a curve in which u1 can be fixed and u2 can be moved is referred to as a u2 curve, a tangent vector along the u1 curve and the u2 curve can be calculated as follows, with a p (u1, u2) point on the curve being a starting point.

<math> <mrow> <mi>e</mi> <mn>1</mn> <mo>=</mo> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>p</mi> </mrow> <msup> <mrow> <mo>&PartialD;</mo> <mi>u</mi> </mrow> <mn>1</mn> </msup> </mfrac> <mo>,</mo> <mi>e</mi> <mn>2</mn> <mo>=</mo> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>p</mi> </mrow> <msup> <mrow> <mo>&PartialD;</mo> <mi>u</mi> </mrow> <mn>2</mn> </msup> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 14)

Then, the unit normal vector n can be calculated from e1, e1 as follows.

<math> <mrow> <mi>n</mi> <mo>=</mo> <mfrac> <mrow> <mi>e</mi> <mn>1</mn> <mo>&times;</mo> <mi>e</mi> <mn>2</mn> </mrow> <mrow> <mo>|</mo> <mo>|</mo> <mi>e</mi> <mn>1</mn> <mo>&times;</mo> <mi>e</mi> <mn>2</mn> <mo>|</mo> <mo>|</mo> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 15)

Thus, three vectors { e1, e2, n } are defined at points on the surface.

At each point, the first elemental quantity E, F, G is defined as follows.

$E={\left|\right|e1\left|\right|}^2,F=(e1,e2),G={\left|\right|\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="A20038010076500171.png,A20038010076500211.png"\; state="\{\{state\}\}">e</figure-callout>}2\left|\right|}^2......$ (equation 16)

Then, the first elementary tensor (g)_ijI, j ═ 1, 2) are defined as follows.

g₁₁＝E，g₁₂＝g₂₁＝F，g₂₂G … … (formula 17)

And, four arrays g_ijI, j ═ 1, 2 is defined as follows.

${\mathrm{<figure-callout\; id="11"\; label="g"\; filenames="A20038010076500171.png"\; state="\{\{state\}\}">g</figure-callout>}}^{11}=\frac{G}{\mathrm{EG}-{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20038010076500171.png,A20038010076500211.png"\; state="\{\{state\}\}">F</figure-callout>}}^2},{\mathrm{<figure-callout\; id="12"\; label="g"\; filenames="A20038010076500171.png"\; state="\{\{state\}\}">g</figure-callout>}}^{12}={\mathrm{<figure-callout\; id="21"\; label="g"\; filenames="A20038010076500171.png"\; state="\{\{state\}\}">g</figure-callout>}}^{21}=\frac{G}{\mathrm{EG}-{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20038010076500171.png,A20038010076500211.png"\; state="\{\{state\}\}">F</figure-callout>}}^2},{\mathrm{<figure-callout\; id="22"\; label="g"\; filenames="A20038010076500171.png"\; state="\{\{state\}\}">g</figure-callout>}}^{22}=\frac{E}{\mathrm{EG}-{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20038010076500171.png,A20038010076500211.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}......$ (equation 18)

Further, at each point, the second basic quantity L, M, N is defined as follows.

<math> <mrow> <mi>L</mi> <mo>=</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&PartialD;</mo> <mn>2</mn> </msup> <mi>p</mi> </mrow> <mrow> <mo>&PartialD;</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mn>1</mn> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&PartialD;</mo> <mn>2</mn> </msup> <mi>p</mi> </mrow> <mrow> <mo>&PartialD;</mo> <msup> <mrow> <msup> <mi>u</mi> <mn>1</mn> </msup> <mo>&PartialD;</mo> <mi>u</mi> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>,</mo> <mi>n</mi> <mo>)</mo> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&PartialD;</mo> <mn>2</mn> </msup> <mi>p</mi> </mrow> <mrow> <mo>&PartialD;</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>,</mo> <mi>n</mi> <mo>)</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 19)

Then, the second elementary tensor (hij, i, j ═ 1, 2) is defined as follows.

h₁₁＝L，h₁₂＝h₂₁＝M，h₂₂Equal to N … … (formula 20)

Now, when the metric { e1, e2, n } is differentiated by the curved surface coordinates (u1, u2), a structural equation of the curved surface represented by the following two equations (the equation of gaussian in equation 21 and the equation of Weigarten in equation 22) is obtained.

<math> <mrow> <mfrac> <msup> <mrow> <mo>&PartialD;</mo> <mi>e</mi> </mrow> <mi>i</mi> </msup> <mrow> <mo>&PartialD;</mo> <msup> <mi>u</mi> <mi>j</mi> </msup> </mrow> </mfrac> <mo>=</mo> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mi>j</mi> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>e</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mi>ij</mi> </msub> <mi>n</mi> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 21)

<math> <mrow> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>n</mi> </mrow> <mrow> <mo>&PartialD;</mo> <msup> <mi>u</mi> <mi>j</mi> </msup> </mrow> </mfrac> <mo>=</mo> <msub> <mrow> <mo>-</mo> <msup> <mi>g</mi> <mi>jk</mi> </msup> <mi>h</mi> </mrow> <mi>ij</mi> </msub> <msub> <mi>e</mi> <mi>k</mi> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 22)

<math> <mrow> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mi>j</mi> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>g</mi> <mi>kl</mi> </msup> <mrow> <mo>(</mo> <mfrac> <msub> <mrow> <mo>&PartialD;</mo> <mi>g</mi> </mrow> <mi>ij</mi> </msub> <msup> <mrow> <mo>&PartialD;</mo> <mi>u</mi> </mrow> <mi>i</mi> </msup> </mfrac> <mo>+</mo> <mfrac> <msub> <mrow> <mo>&PartialD;</mo> <mi>g</mi> </mrow> <mi>li</mi> </msub> <msup> <mrow> <mo>&PartialD;</mo> <mi>u</mi> </mrow> <mi>j</mi> </msup> </mfrac> <mo>+</mo> <mfrac> <msub> <mrow> <mo>&PartialD;</mo> <mi>g</mi> </mrow> <mi>ij</mi> </msub> <msup> <mrow> <mo>&PartialD;</mo> <mi>u</mi> </mrow> <mi>l</mi> </msup> </mfrac> <mo>)</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 23)

Equation 23 represents the Christoffel symbol.

The integratable condition for constructing equations 21, 22 is expressed in the following two equations (gaussian equation of equation 24 and meilnidi-kodachi equation of equation 25).

$R_{\mathrm{jkl}}^i=g^{\mathrm{im}}(h_{\mathrm{jk}}{h}_{\mathrm{lm}}-h_{\mathrm{jl}}{h}_{\mathrm{km}})......$ (equation 24)

<math> <mrow> <mfrac> <mrow> <mo>&PartialD;</mo> <msub> <mi>h</mi> <mi>ij</mi> </msub> </mrow> <mrow> <mo>&PartialD;</mo> <msup> <mi>u</mi> <mi>k</mi> </msup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mo>&PartialD;</mo> <msub> <mi>h</mi> <mi>ik</mi> </msub> </mrow> <mrow> <mo>&PartialD;</mo> <msup> <mi>u</mi> <mi>j</mi> </msup> </mrow> </mfrac> <mo>+</mo> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> <mi>l</mi> </mtd> <mtd> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mi>j</mi> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>h</mi> <mi>lk</mi> </msub> <mo>-</mo> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>l</mi> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>h</mi> <mi>ij</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 25)

<math> <mrow> <msubsup> <mi>R</mi> <mi>jkl</mi> <mi>i</mi> </msubsup> <mo>=</mo> <mfrac> <mo>&PartialD;</mo> <mrow> <mo>&PartialD;</mo> <msup> <mi>u</mi> <mi>l</mi> </msup> </mrow> </mfrac> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mfrac> <mo>&PartialD;</mo> <mrow> <mo>&PartialD;</mo> <msup> <mi>u</mi> <mi>k</mi> </msup> </mrow> </mfrac> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> <mtd> <mi>l</mi> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> </mrow> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mfenced> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mi>l</mi> </mtd> </mtr> </mtable> </mfenced><mo>-</mo> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> <mtd> <mi>l</mi> </mtd> </mtr> </mtable> </mfenced> <mfenced open='{' close='}'> <mtable> <mtr> <mtd> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mfenced> </math> (equation 26)

Where equation 26 represents the curvature tensor of Riemancherestofol.

The first elementary tensor (gij, i, j ═ 1, 2) and the second elementary tensor (hij, i, j ═ 1, 2) are provided as a function of the coordinates (u1, u2) of the curved surface, and in the case where they satisfy the above gaussian equation and the meineldi-kodachi equation, the shape of the curved surface having such gij, hij is uniquely determined (refer to the basic theorem of polynary naphthalene (ボネ)), so that the respective meshes C2 are continuous.

The CPU performs these arithmetic processes and calculates the integratable conditions (step S2).

Next, the CPU executes curvature line analysis processing, feature line analysis processing, and curvature/circumference conversion processing (step S3). First, main curvatures κ 1 and κ 2 in the mesh are calculated based on the primary specification E, F, G and the secondary specification L, M, N by curvature line analysis processing (step S4). That is, the extreme value of the curvature κ is first calculated. The shape of the normal cross section, which is the intersection of the normal plane and the curved surface shown in fig. 3, changes simultaneously with the tangential direction, and the normal curvature changes accordingly. This shape returns to the original shape when the normal plane is half-rotated. Now, let γ be

<math> <mrow> <mi>&gamma;</mi> <mo>=</mo> <mfrac> <mi>dv</mi> <mi>du</mi> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 27)

Further, when κ is rewritten to γ, the function κ (γ) becomes { L- κ (γ) · E } +2{ M- κ (γ) · F } γ + { N- κ (γ) · G } γ²0 … … (equation 28). According to the quadratic formula of γ, κ (γ) assumes an extreme value in d κ (γ)/d γ ═ 0. Then, when equation 15 is differentiated based on the extreme value condition and κ and γ are rewritten into (κ -) and (γ -), the equation is obtained

<math> <mrow> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>F</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mi>N</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>G</mi> <mo>)</mo> </mrow> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 29).

Then, when the result is substituted into equation 16, the result is obtained

<math> <mrow> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>E</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mo>)</mo> </mrow> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 30).

From these equations, the following relational expressions are obtained.

<math> <mrow> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>F</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mi>N</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>G</mi> <mo>)</mo> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>L</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>F</mi> <mo>)</mo> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 31)

<math> <mrow> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mi>M</mi> <mo>+</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>N</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>F</mi> <mo>+</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>G</mi> <mo>)</mo> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mi>L</mi> <mo>+</mo> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mi>M</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>E</mi> <mo>-</mo> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mi>F</mi> <mo>)</mo> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 32)

When equation 18 is modified, the result is

<math> <mrow> <mrow> <mo>(</mo> <mi>EG</mi> <mo>-</mo> <msup> <mi>F</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <msup> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mn>2</mn> </msup> <mo>-</mo> <mrow> <mo>(</mo> <mi>EN</mi> <mo>+</mo> <mi>LG</mi> <mo>-</mo> <mn>2</mn> <mi>MF</mi> <mo>)</mo> </mrow> <mover> <mi>&kappa;</mi> <mo>~</mo> </mover> <mo>+</mo> <mi>LN</mi> <mo>-</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 33).

When the coefficient of κ to 2 is positive and the root is κ 1 and κ 2 according to equation 7, the value becomes the principal curvature as shown in fig. 5.

Next, a gaussian curvature or an average curvature is calculated based on the principal curvature (step S5). That is, the relationship between the root of the quadratic equation and the coefficient is expressed as

<math> <mrow> <mi>Km</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&kappa;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&kappa;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfrac> <mrow> <mo>(</mo> <mi>EN</mi> <mo>+</mo> <mi>LG</mi> <mo>-</mo> <mn>2</mn> <mi>MF</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>EG</mi> <mo>-</mo> <msup> <mi>F</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 34)

<math> <mrow> <mi>Kg</mi> <mo>=</mo> <msub> <mi>&kappa;</mi> <mn>1</mn> </msub> <msub> <mi>&kappa;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mi>LN</mi> <mo>-</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>EG</mi> <mo>-</mo> <msup> <mi>F</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 35).

Here, Km and Kg are the average curvature and gaussian curvature, respectively. As shown in fig. 6, when Kg — 0 is a curved surface, the curved surface becomes an expandable surface, and the curvature line on the curved surface becomes a straight line. In the present embodiment, the point at which the gaussian curvature is 0 is set as a reference point of deformation described later.

In addition to the points, examples of suitable points as the reference points for the deformation include curvature lines, boundary lines (ridge lines), equi-oblique orthogonal lines shown in fig. 7, principal curvature extreme lines shown in fig. 8, oblique extreme lines shown in fig. 9, and umbilical points. These are points or lines serving as reference points or reference lines of deformation defined by a change pattern of one or two or more characteristic quantities among a principal curvature, a principal direction, a gaussian curvature, an average curvature, and a curvature line, which are characteristic quantities representing characteristics of a curved surface, and can be calculated based on the primary standard quantity and the secondary standard quantity.

Furthermore, curvature lines representing the principal directions of the mesh are calculated based on the principal curvatures. That is, when κ is deleted from equation 19, it is obtained

<math> <mrow> <mrow> <mo>(</mo> <mi>MG</mi> <mo>-</mo> <mi>NF</mi> <mo>)</mo> </mrow> <msup> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mi>GL</mi> <mo>-</mo> <mi>NE</mi> <mo>)</mo> </mrow> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mo>+</mo> <mi>FL</mi> <mo>-</mo> <mi>ME</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 36)

Or,

(MG-NF)dv²+(GL-NE)dudv+(FL-ME)du²0 … … (equation 37).

Both of these equations are equations of curvature lines, and γ 1 and γ 2 have the following relationship because they are quadratic equations.

<math> <mrow> <msub> <mi>&gamma;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&gamma;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>GL</mi> <mo>-</mo> <mi>NE</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mi>MG</mi> <mo>-</mo> <mi>NF</mi> <mo>)</mo> </mrow> </mfrac> <mo>,</mo> <msub> <mi>&gamma;</mi> <mn>1</mn> </msub> <msub> <mi>&gamma;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mi>FL</mi> <mo>-</mo> <mi>ME</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>MG</mi> <mo>-</mo> <mi>NF</mi> <mo>)</mo> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math> (equation 38)

At a point on the curved surface, the curvature assumes an extreme value in the direction determined by γ 1 and γ 2. The tangent vector on the curved surface is (Sudu + Svdv), and the inner product of the two tangent vectors corresponding to γ 1 and γ 2 is

(ds)₁·(ds)₂＝{(s_u+s_vγ₁)·(s_u+s_vγ₂)}(du)₁(dv)₂… … (calculator 39)

When the { } is transformed internally,

{ E (MG-NF) -F (GL-NE) + G (FL-ME) }/(MG-NF) … … (equation 40) becomes zero. That is, it is known that two tangential directions of the normal section of the principal curvature are orthogonal to each other. This direction is referred to as a principal direction, and when the principal direction coincides with a tangent on the curved surface, this becomes a curvature line shown in fig. 10.

As described above, the curvature line calculation process representing the main direction of the mesh is performed.

Next, a curve circumferential length conversion process is executed (step S6). That is, the CPU calculates the circumference based on the curvature calculated from the primary specification amount E, F, G and the secondary specification amount L, M, N. Along the curvature line calculated by the above-described calculation processing of the curvature line, the curvature radius r is calculated from the curvature (1/r), and the circumference of the curvature line is expanded and contracted every calculation section.

Thus, the analysis process is performed.

Next, the CPU receives the collection of the point row information created and extracted in step S1 and step S2, and the primary and secondary specifications (yes in step S7), and performs the curved surface data transfer process (step S9). On the other hand, if these pieces of information are not complete, the database evaluation process is performed (no in step S7). That is, when the shape reproduced based on the main direction, the reference position (point, line, etc.), and the deformation amount calculated in steps S4 to S6 and the shape reproduced based on the point row information and the primary and secondary specifications are compared and matched (yes in step S8), the curved surface data is transferred (step S9). If they do not match (no in step S8), the approximate complementary accuracy improving process is performed. That is, the original curved surface is approximately supplemented so as to be differentiable in the second order, and the above-described processing is repeated again from step S1. Then, when the comparison and evaluation in step S8 match, the process proceeds to the data transfer process.

The curved surface data is transferred to the conversion program 2 or the reproduction program 3 shown in fig. 1. The CPU executes the conversion program 2 upon receiving the conversion command. That is, first, a point having a gaussian curvature of 0 acquired as a feature point or a feature line is used as a deformation reference, and a curvature line is deformed by stretching only a peripheral length portion in a curvature line direction, thereby reproducing a mesh or a curved surface. Then, a plurality of point rows on the curved surface are extracted from the reproduced mesh or curved surface, and the point rows are transformed according to a graphic representation algorithm in other CAD systems. The converted graphic data is reproduced by another CAD application 22 and then output to the image display processing unit 11. The image display processing unit 11 performs display processing of data output from the CAD application 22, and outputs the data to the display unit 12. The display unit 12 receives input of display data and displays the display data.

Then, the CPU executes the reproduction program 3 when receiving a reproduction command. The reproduction program causes the CPU to execute processing other than the conversion processing in the conversion program. That is, a point having a gaussian curvature of 0 is used as a reference for deformation, and only the curvature line is deformed by stretching in the curvature line direction by the circumferential length, thereby reproducing a mesh or a curved surface. The reproduced graphics data is then output to the image display processing unit 11, and after the display processing, is displayed on the display unit 12.

As described above, according to the CAD system of the present embodiment, the effect of enabling analysis, conversion, and reproduction of a free-form surface is obtained while maintaining the continuity of C2. Thus, the utility value of the CAD model can be greatly improved, and the effect of making the design and production process efficient can be obtained.

In the CAD system according to the present embodiment, a series of processes for analyzing, converting, and reproducing a free-form surface in a CAD model has been described, but the CAD system according to the present invention is not limited to this, and can be applied to a system and a program for performing image representation using the CG system and a computer.

In the CAD system according to the embodiment, as a suitable example, as shown in fig. 2, a free-form surface is analyzed, converted, and reproduced by dividing the surface into meshes, normalizing the meshes with the basis vectors Su and Sv, and expressing u and v parameters using the point sequence information (u, v).

The CAD system has a computer system therein. The series of processes relating to the analysis, conversion, and reproduction of the free-form surface are stored in a computer-readable recording medium in the form of a program, and the processes are executed by reading the program and executing it by a computer. The term "computer-readable recording medium" as used herein refers to magnetic disks, optical disks, CD-ROMs, DVD-ROMs, semiconductor memories, and the like. The computer program may be distributed to the computers via a communication line, and the computer that has received the distribution may execute the program.

Claims (10)

1. A CAD system, comprising:

a point row information extraction section that extracts a plurality of point rows on the curved surface;

a dividing means for generating a curved surface from the point sequence by using another CAD system and dividing the curved surface into a predetermined number of meshes;

a primary specification amount calculation unit for calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh;

a secondary specification amount calculation unit that calculates a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and

and a storage unit that stores the dot row information, the primary specification amount, and the secondary specification amount.

2. The CAD system of claim 1, further comprising:

a principal curvature calculation unit that calculates a principal curvature of the mesh based on the primary and secondary specification amounts;

a curvature line calculation unit that calculates a curvature line representing a principal direction of the mesh based on the principal curvature;

a feature point/feature line analyzing unit that extracts a point or a line serving as a reference point or a reference line of a deformation defined by a variation pattern of one or two or more feature amounts from five feature amounts indicating a feature of the curved surface, the five feature amounts being composed of a gaussian curvature and an average curvature calculated based on the principal curvature, the principal direction, the curvature line, and the primary and secondary specification amounts; and

and a circumferential length calculating unit that calculates a circumferential length based on the curvature calculated from the primary and secondary specification amounts.

3. The CAD system of claim 2, further comprising:

and a reproduction unit that reproduces the mesh or the curved surface by deforming only the curvature line of the circumferential portion in the curvature line direction with the feature point or the feature line as a reference for deformation.

4. The CAD system of claim 3, further comprising:

and a conversion unit that extracts a plurality of point sequences on the curved surface from the reproduced mesh or curved surface and converts the point sequences according to a graphic expression algorithm in another CAD system.

5. A CAD program characterized by causing a computer to execute:

extracting point column information, namely extracting a plurality of point columns on the curved surface;

a dividing process of generating a curved surface by using another CAD system based on the point sequence and dividing the curved surface into a predetermined number of meshes;

a primary specification amount calculation process of calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh;

a secondary specification amount calculation process of calculating a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and

and a storing process of storing the dot column information, the primary specification amount, and the secondary specification amount.

6. The CAD program of claim 5, further causing the computer to execute:

a principal curvature calculation process of calculating a principal curvature of the mesh based on the primary and secondary specification amounts;

a curvature line calculation process of calculating a curvature line representing a principal direction of the mesh based on the principal curvature;

a feature point/feature line analysis process of extracting a point or a line serving as a reference point or a reference line of a deformation defined by a variation pattern of one or two or more feature amounts, from five feature amounts representing features of the curved surface, the five feature amounts being composed of the principal curvature, the principal direction, the curvature line, and the gaussian curvature and the average curvature calculated from the primary and secondary specification amounts; and

and a perimeter calculation process of calculating a perimeter based on the curvature calculated from the primary and secondary specification amounts.

7. The CAD program of claim 6, further causing the computer to:

and a reproduction process of deforming only the curvature line of the peripheral portion in the curvature line direction with the feature point or the feature line as a reference of deformation, and reproducing the mesh or the curved surface.

8. The CAD program of claim 7, further causing the computer to:

and a transformation process of extracting a plurality of point rows on the curved surface from the reproduced mesh or curved surface and transforming the point rows according to a graphic representation algorithm in other CAD systems.

9. A CG system, comprising:

a point row information extraction section that extracts a plurality of point rows on the curved surface;

a dividing means for generating a curved surface from the point sequence by using another CG system and dividing the curved surface into a predetermined number of meshes;

a primary specification amount calculation unit for calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh;

a secondary specification amount calculation unit that calculates a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and

and a storage unit that stores the dot row information, the primary specification amount, and the secondary specification amount.

10. A CG program for causing a computer to execute:

extracting point column information, namely extracting a plurality of point columns on the curved surface;

a division process of generating a curved surface using another CG system based on the point sequence and dividing the curved surface into a predetermined number of meshes;

a primary specification amount calculation process of calculating a primary specification amount defined by a tangent vector of a tangent plane forming the mesh;

a secondary specification amount calculation process of calculating a secondary specification amount defined by the tangent vector and a normal vector of the mesh; and

and a storing process of storing the dot column information, the primary specification amount, and the secondary specification amount.

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